## Abstract

Consider an (L, α)-superdiffusion X on ℝ^{d}, where L is an uniformly elliptic differential operator in ℝ^{d}, and 1 < α < 2. The G-polar sets for X are subsets of ℝ × ℝ^{d} which have no intersection with the graph G of X, and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the G-polarity of a general analytic set A ⊂ ℝ × ℝ^{d} in term of the Bessel capacity of A, and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the G-polarity of sets of the form E × F, where E and F are two Borel subsets of ℝ and ℝ^{d} respectively. We establish a relationship between the restricted Hausdorff dimension of E × F and the usual Hausdorff dimensions of E and F. As an application, we obtain a criterion for G-polarity of E × F in terms of the Hausdorff dimensions of E and F, which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.

Original language | English |
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Pages (from-to) | 3721-3728 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 127 |

Issue number | 12 |

State | Published - 1 Dec 1999 |

## Keywords

- Box dimension
- G-polarity
- Graph of superdiffusion, semilinear partial differential equation
- H-polarity
- Hausdorff dimension
- Restricted hausdorff dimension
- Superdiffusion